The Banach ideal of A-compact operators and related approximation properties

Abstract

Abstract. We use the notion of A-compact sets (determined by an operator ideal A), introduced by Carl and Stephani (1984), to show that many known results of certain approximation properties and several ideals of compact operators can be systematically studied under this framework. For Banach operator ideals A, we introduce a way to measure the size of A-compact sets and use it to give a norm on K A , the ideal of A-compact operators. Then, we study two types of approximation properties determined by A-compact sets. We focus our attention on an approximation property which makes use of the norm defined on K A . This notion fits the definition of the A-approximation property, recently introduced by Oja (2012), with K A instead of A. We exemplify the power of the Carl-Stephani theory and the geometric structure introduced here by appealing to some recent developments on p-compactness.

Ideal topologies and corresponding approximation properties

“…Proposition 2.4 can be used to provide many further useful examples of ideal topologies. Given an operator ideal I and a Banach space E, according to [63,28,37] we define…”

“…The consideration of problems on the approximation by operators belonging to a given operator ideal was a question of time. Indeed, a number of approximation properties (APs) with respect to operator ideals-and other ones that are somehow related to operator ideals-have been studied in the last three decades, see, e.g., [6,11,13,15,19,20,29,34,35,37,38,39,40,41,49,50,57,58,59,62,64]. The reader is also referred to the surveys [51,52] and to the references therein.…”

$${\mathcal A}$$ A -compact mappings

“…A sequence (x n ) n ⊂ X is A-null if there exists a Banach space Z, an operator T ∈ A(Z; X) and a null sequence (z n ) n ⊂ Z such that x n = T z n for all n ∈ N. Also, from [19, Proposition 1.4] we have that a sequence (x n ) n ⊂ X is A-null if and only if, (x n ) n is relatively A-compact and norm convergent to zero. The size of a relatively A-compact set is defined in [19] as follows. For a relatively A-compact set K ⊂ X,…”

“…The space of all A-compact operators from X to Y is denoted by K A . This space becomes a Banach operator ideal we endow it with the norm (see [19])…”

“…Given a (Banach) operator ideal A, a set K of a Banach space X is relatively A-compact if there exist a Banach space Z, a compact set L ⊂ Z and a linear operator T ∈ A(Z; X) such that K ⊂ T (L). Sinha and Karn [24] defined, for 1 ≤ p < ∞, relatively p-compact sets which, by [19], coincides with relatively N p -compact sets. Here N p stands for the ideal of right p-nuclear operators, see for instance [22, p.140] for the definition of this ideal.…”